Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Use the Fixed-Point Algorithm with \(x_{1}\) as indicated to solve the equations to five decimal places. $$ x=2-\sin x ; x_{1}=2 $$

Step-by-Step Solution

Answer

The solution to five decimal places is approximately 1.15221.

Step 1: Understanding the Fixed-Point Algorithm

The Fixed-Point algorithm iteratively aims to find a solution to the equation \(x = g(x)\), where \(g(x)\) is a function derived from rearranging the original equation. We repeatedly substitute the current approximation into \(g(x)\) to obtain the next approximation. The process is continued until the change between successive approximations is less than a specified tolerance.

Step 2: Rearrange the Equation

We start with the equation \(x = 2 - \sin x\). Here, \(g(x) = 2 - \sin x\). This function will be repeatedly evaluated to find the fixed point solution.

Step 3: Initial Approximation

We have been given the initial approximation \(x_1 = 2\). This is our starting point for the iteration.

Step 4: Perform the Iterations

Using the formula \(x_{n+1} = g(x_n) = 2 - \sin(x_n)\), perform successive iterations:- Iteration 1: \ x_2 = 2 - \sin(2) \approx 1.0907026- Iteration 2: \ x_3 = 2 - \sin(1.0907026) \approx 1.1605688- Iteration 3: \ x_4 = 2 - \sin(1.1605688) \approx 1.1505243- Iteration 4: \ x_5 = 2 - \sin(1.1505243) \approx 1.1525358- Iteration 5: \ x_6 = 2 - \sin(1.1525358) \approx 1.1522106Continue until the change between successive iterations is below the desired precision.

Step 5: Check for Convergence

Examine the difference between successive approximations. The iteration appears to be converging as the difference between \(x_5\) and \(x_6\) is less than 0.00001, meeting the precision requirement of five decimal places.

Step 6: Conclude the Solution

With convergence confirmed, the approximate fixed-point solution to five decimal places is \(x \approx 1.15221\).

Key Concepts

Iteration Method

The iteration method is a powerful tool in mathematics used to refine approximate solutions for equations. It's like repeatedly trying corrections to inch closer to a solution. The approach involves starting with an initial guess and then using a specific formula to generate subsequent approximations.

In the context of the Fixed-Point Algorithm, this process can be visualized as feeding an initial value into a function, getting an output, and then using that output as the new input for the next round. It's like a loop where each cycle refines the previous answer. This is particularly helpful for solving equations that are hard to crack algebraically.

The iteration method doesn't always guarantee a solution. However, with a well-chosen starting point and under certain conditions, it often leads to a satisfactory and precise solution.

Numerical Analysis

Numerical analysis is all about using numbers and computations to understand and solve mathematical problems. Unlike standard algebra, which relies on symbolic manipulation, numerical analysis focuses on algorithms and approximation techniques to tackle complex equations.

This field of study is crucial as it allows us to handle real-world problems where symbolic solutions are not feasible. For example, when dealing with trigonometric functions like sine or cosine, exact answers aren't always easy to find, and that's where numerical methods step in.

By employing techniques like the Fixed-Point Algorithm, numerical analysis offers a way to systematically improve guesses through iteration. This helps in approaching the true solution gradually, even for challenging equations.

Convergence

In the realm of numerical methods, convergence is the key measure of success. It refers to the process of approaching a final, accurate solution through a sequence of iterations or approximations.

For the Fixed-Point Algorithm, convergence is achieved when the difference between successive approximations becomes smaller than a predefined threshold, indicating that we're honing in on the correct value.

It's important because without convergence, iterations might continue endlessly or oscillate without stabilizing. For this reason, checking the difference between iterations is critical. This ensures the process is both efficient and productive, leading to reliable outcomes.

Approximation Techniques

Approximation techniques are the backbone of numerical solutions, providing a practical way to solve equations that lack exact answers. These techniques allow us to find a near-accurate solution by estimating the values incrementally.

In the case of the Fixed-Point Algorithm, we use an initial guess and then continuously refine it through repeated evaluations of a function. This approach is particularly useful when the exact solution is either unknown or impossible to acquire analytically.

These techniques shine because they balance efficiency with precision—getting us close enough to the true solution for practical purposes without requiring excessive computational resources or time. This makes them invaluable in fields ranging from engineering to computational sciences.